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Question Number 77842 by jagoll last updated on 11/Jan/20

$$given$$$$f\left(x\right)=x^2+\sin \left(2x\right)+\int \underset{0}{\stackrel{\frac{\pi }{4}}{}}f\left(x\right)dx$$$$f\left(\frac{\pi }{2}\right)=?$$

Answered by mr W last updated on 11/Jan/20

$$let{\int }_0^{\frac{\pi }{4}}f\left(x\right)dx=a=constant$$$$f\left(x\right)=x^2+\sin \left(2x\right)+a$$$${\int }_0^{\frac{\pi }{4}}f\left(x\right)dx={[\frac{x^3}{3}-\frac{\cos \left(2x\right)}{2}+ax]}_0^{\frac{\pi }{4}}=a$$$$\frac{{\pi }^3}{192}+\frac{1}{2}+\frac{a\pi }{4}=a$$$$\Rightarrow a=\frac{{\pi }^3+96}{48(4-\pi )}$$$$f\left(x\right)=x^2+\sin \left(2x\right)+\frac{{\pi }^3+96}{48(4-\pi )}$$$$f\left(\frac{\pi }{2}\right)=\frac{{\pi }^2}{4}+\frac{{\pi }^3+96}{48(4-\pi )}=\frac{(48-11\pi ){\pi }^2+96}{48(4-\pi )}$$

Commented by jagoll last updated on 11/Jan/20

$$thanksyousir$$

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